Integrand size = 13, antiderivative size = 25 \[ \int \frac {\text {sech}^2(x)}{i+\sinh (x)} \, dx=-\frac {i \text {sech}(x)}{3 (i+\sinh (x))}-\frac {2}{3} i \tanh (x) \]
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Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2751, 3852, 8} \[ \int \frac {\text {sech}^2(x)}{i+\sinh (x)} \, dx=-\frac {2}{3} i \tanh (x)-\frac {i \text {sech}(x)}{3 (\sinh (x)+i)} \]
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Rule 8
Rule 2751
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {sech}(x)}{3 (i+\sinh (x))}-\frac {2}{3} i \int \text {sech}^2(x) \, dx \\ & = -\frac {i \text {sech}(x)}{3 (i+\sinh (x))}+\frac {2}{3} \text {Subst}(\int 1 \, dx,x,-i \tanh (x)) \\ & = -\frac {i \text {sech}(x)}{3 (i+\sinh (x))}-\frac {2}{3} i \tanh (x) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\text {sech}^2(x)}{i+\sinh (x)} \, dx=-\frac {1}{3} i \left (\frac {\text {sech}(x)}{i+\sinh (x)}+2 \tanh (x)\right ) \]
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Time = 27.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-\frac {4 \left (2 \,{\mathrm e}^{x}+i\right )}{3 \left ({\mathrm e}^{x}+i\right )^{3} \left ({\mathrm e}^{x}-i\right )}\) | \(24\) |
default | \(-\frac {i}{2 \left (-i+\tanh \left (\frac {x}{2}\right )\right )}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {2 i}{3 \left (\tanh \left (\frac {x}{2}\right )+i\right )^{3}}-\frac {3 i}{2 \left (\tanh \left (\frac {x}{2}\right )+i\right )}\) | \(49\) |
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none
Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {\text {sech}^2(x)}{i+\sinh (x)} \, dx=-\frac {4 \, {\left (2 \, e^{x} + i\right )}}{3 \, {\left (e^{\left (4 \, x\right )} + 2 i \, e^{\left (3 \, x\right )} + 2 i \, e^{x} - 1\right )}} \]
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\[ \int \frac {\text {sech}^2(x)}{i+\sinh (x)} \, dx=\int \frac {\operatorname {sech}^{2}{\left (x \right )}}{\sinh {\left (x \right )} + i}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (15) = 30\).
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {\text {sech}^2(x)}{i+\sinh (x)} \, dx=-\frac {8 \, e^{\left (-x\right )}}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} + \frac {4 i}{-6 i \, e^{\left (-x\right )} - 6 i \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} - 3} \]
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {\text {sech}^2(x)}{i+\sinh (x)} \, dx=\frac {1}{2 \, {\left (e^{x} - i\right )}} - \frac {3 \, e^{\left (2 \, x\right )} + 12 i \, e^{x} - 5}{6 \, {\left (e^{x} + i\right )}^{3}} \]
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Time = 1.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {\text {sech}^2(x)}{i+\sinh (x)} \, dx=-\frac {8\,{\mathrm {e}}^x}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}-\frac {8\,{\mathrm {e}}^x\,\left ({\mathrm {e}}^{2\,x}-1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}+\frac {{\mathrm {e}}^{2\,x}\,16{}\mathrm {i}}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3}-\frac {\left ({\mathrm {e}}^{2\,x}-1\right )\,4{}\mathrm {i}}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \]
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